What are some profound scientific or mathematical truths
>>16454471Reality is bigger than rationality, formally and colloquially
>>16454471https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem
>>16454471/sci/ is for midwits
>>16454471The state of a system of many things that can interact in many different ways, will on average require more information to describe than what it started with, because there are more states that require more information to specify than states that require less information to specify.
>>16454603Literally how do you even understand its implications without having a PhD in maths?
>>16454476What does this even mean? Be honest you just put a bunch of big words together to make you sound smart.
>>16454730The Grothendieck–Riemann–Roch (GRR) theorem has profound implications in several areas of mathematics, particularly algebraic geometry, topology, and complex analysis. Here are some of the key implications:Bridge Between Geometry and Algebraic Topology: The GRR theorem connects algebraic and topological invariants of coherent sheaves on complex manifolds, enabling complex analytic and algebraic information (like sheaf cohomology) to be translated into topological data (like Chern classes and characteristic classes). This creates a pathway between algebraic geometry and topological K-theory, broadening the scope for deeper insights in both areas.Foundation for Index Theory: The GRR theorem serves as a conceptual foundation for the Atiyah-Singer Index Theorem, which calculates the index of elliptic operators on compact manifolds. By relating the Euler characteristic (a topological invariant) of sheaves with integrals of Chern classes, GRR provides tools to handle complex calculations in index theory and differential geometry.Tool for Computing Intersection Numbers: GRR allows for the computation of intersection numbers and dimensions in a more straightforward manner through the use of Chern classes and K-theory. This has practical applications in enumerative geometry, where the aim is often to count the number of solutions to geometric problems.Unified Framework for Riemann–Roch Type Theorems: The classical Riemann–Roch theorem is a specific case of the broader GRR framework. By generalizing Riemann–Roch to higher-dimensional varieties and more general morphisms, it gives a unified theory for computing Euler characteristics across different contexts, from curves to complex varieties.These implications make the GRR theorem a central result in modern geometry and have led to significant advances in the study of moduli spaces, mirror symmetry, and string theory, where connections between geometry, physics, and topology are explored.
a+b=b+a
>>16455164Fuck this ChatGPT output
>>16454603>https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theoremi hope to understand what the hell any of this means one day
>>16454716That sounds like Gödel's incompleteness theorem with extra steps.
>>16454471The profoundedest thing is that none of it is all that profound, no matter how deep you go. For instance, perhaps nature is le quantum fields? So what? It makes no difference on a philosophical level. Physics cannot at this time differentiate between QM interpretations, but that could be a little more interesting.
